Spectral efficiency is of great importance in current and future wireless communications. This has led to a move toward using the high order constellations. Orthogonal Frequency Division Multiplex (OFDM) is a well-established technology used in current and the next generation of wireless transmission links.
In coherent digital communication, high frequency carrier waveforms are used for the modulation and the demodulation of desired signals. A local oscillator (LO) at the transmitter side is used to generate the carrier signal as part of the modulation process. Conversely, a similar local oscillator is used in the demodulation process at the receiver side. In almost all practical LOs, the generated carrier waveform contains an unwanted phase offset, which is known as the oscillator phase noise (PN). Lower the PN of the carrier can only be achieved using high price the oscillators. It is common to use inexpensive LOs at both transmit and receive ends of the transmission, while compensating for the PN effect through a digital signal processing (DSP) at the receiver.
Orthogonal frequency division multiplexing (OFDM) is a common approach to battle the multipath effect in mobile communication systems. The OFDM is currently deployed in long term evolution (LTE) networks and it is proposed to be used in the 5th generation new radio (5G-NR) as well. OFDM uses large modulation constellations, such as for example the 256-quadrature amplitude modulation (QAM). Such large constellations benefit from high spectral efficiencies but also require high signal to noise ratio (SNR) levels.
The performance of the OFDM systems is severely deteriorated in the presence of PN. In particular, PN affects the OFDM systems by causing common phase error (CPE) and intercarrier interference (ICI). CPE is a joint effect of PN on all subcarriers (SC) within the OFDM symbol. CPE causes the rotation of the whole constellation. ICI causes a loss of orthogonality between the SCs. An effect of ICI appears as a set of additive terms added to the desired signal at the output of the OFDM demodulator.
FIG. 1 is a graph showing an example of an OFDM system using 256-QAM constellation affected by common phase error and intercarrier interference. FIG. 2 is a graph showing the OFDM system using 256-QAM constellation of FIG. 1 after removal of the effect of common phase error effect. On FIGS. 1 and 2, it is assumed that the radio channel between the transmitter and the receiver is mainly subject to additive white Gaussian noise (AWGN).
The PN effect visible in FIGS. 1 and 2 is caused by imperfect LOs, both at the transmitter and the receiver sides. Regardless of the constellation size, the PN effect must be compensated at the receiver in order to prevent the output SNR from becoming extremely low, sometimes negative.
Due to the nature of the OFDM technology, the transmission link operates in the frequency domain (FD) before the OFDM modulation, in the time domain (TD) over the channel, and again in FD after the OFDM demodulation. PN suppression methods may thus be implemented in either of FD or TD.
Three (3) main factors should be considered when evaluating the performance and cost of various PN suppression methods. One such factor is computational complexity. A number of complex multiplications and additions during the estimation process, a dimension of matrices that need to be inversed, and any other considerable computations needs to be taken into account. Another factor is the processing delay, which is of utmost importance in the context of 5G-NR. Finally, pilot overhead, which may be defined as a ratio between a number of transmitted pilot symbols and a number of SCs within one OFDM symbol, should be limited in order to maintain the expected spectral efficiency.
Frequency Domain PN Suppression Method
The effect of PN on the transmitted symbols at the output of the demodulator reflects as CPE and ICI. The relationship between the OFDM demodulator output and the transmitted constellation symbols at a given subcarrier (SC) is expressed in equation (1):
                              Y          ⁡                      (            k            )                          =                                            ∑                              m                =                0                                            N                -                1                                      ⁢                                          R                ⁡                                  (                                                            [                                              k                        -                        m                                            ]                                        N                                    )                                            ·                              I                ⁡                                  (                  m                  )                                                              =                                                                      R                  ⁡                                      (                    k                    )                                                  ⁢                                                                            I                      ⁡                                              (                        0                        )                                                              ︸                                    CPE                                            +                                                                                                                  ∑                                                  m                          =                          1                                                                          N                          -                          1                                                                    ⁢                                                                        R                          ⁡                                                      (                                                                                          [                                                                  k                                  -                                  m                                                                ]                                                            N                                                        )                                                                          ⁢                                                  I                          ⁡                                                      (                            m                            )                                                                                                                ︸                                    ICI                                ⁢                0                                      ≤            k            ≤                          N              -              1                                                          (        1        )            
In equation (1), R(k)=S (k)H(k), where S and H are the transmitted constellation symbol and the frequency response of the channel at the k'th SC respectively, [q]N shows q modulo N operation, the parameter I which contains the effect of PN is defined as
            I      ⁡              (        k        )              =                  1        N            ⁢      FFT      ⁢              {                  e                      j            ⁢                                                  ⁢            ϕ                          }              ,where ϕ=ϕtx+ϕrx and FFT{A} is a N point Fast Fourier Transform (FFT) operation of variable A. Equation (1) reveals that the CPE effect is independent of the SC index, while ICI is different at each SC.
In the FD PN suppression method, a DSP algorithm is implemented in the receiver, after the OFDM demodulation. The FD PN suppression is performed in two stages. In a first stage, CPE is removed through an averaging procedure. The computational complexity and the required overhead of this first stage are small. However, removing the CPE alone is only sufficient in the case of small constellations, such as quadrature phase shift keying (QPSK) or 16-QAM. To further suppress the PN effect from OFDM demodulated signals, the effect of ICI needs to be mitigated as well. As opposed to the CPE removal, ICI mitigation is a complex process. This process not only requires more known pilot symbols among transmitted data symbols, it also requires an algorithm capable of estimating ICI in every SC, using the knowledge of the pilot symbols.
The following general ICI mitigation approach is followed in conventional FD PN suppression methods. Their mathematical models are based on the assumption that PN is a low frequency process, in the sense that a power spectral density (PSD) of the PN is mainly concentrated in low frequency components. With this assumption, the ICI mitigation problem is formulated as an estimation of a few small frequency components and neglecting the rest. Otherwise stated, instead of finding all of the N ICI terms in equation (1), the FD PN suppression method approach only estimates 2l+1 terms, in which l is usually less than 5 among the N ICI terms in equation (1). Adding 1 is attributed to the CPE removal, which corresponds to SC 0 (a null SC position). Several mathematical approaches may be adopted to estimate the aforementioned 2l+1 components from the received pilot information. In terms of computational complexity, the solution to the estimation problem, in its simplest form, includes a matrix inversion of dimension (2l+1)×(2l+1). This matrix inversion forms a major computational burden in the FD PN suppression method.
Among various FD PN suppression methods, an iterative approach has been proposed to further improve the initial estimation result. A particular algorithm for this iterative approach includes a feedback loop that uses a detected data symbols after using a forward error correction (FEC) process. In each iteration, an estimation problem is re-solved and the 2l+1 ICI terms are re-estimated. The difference in each iteration lies in a formed set of equations. As more reliable data symbols become available at the output of the FEC, the equation coefficients are updated. As a result, a better estimate of ICI terms may be achieved by solving the updated equation. The estimated ICI terms, obtained using this iterative scheme, are more accurate when compared to the one time solving of the equations.
FD PN suppression methods, whether it is iterative or not, are effective in compensating the effect of PN when the constellation size is 64-QAM or smaller. However, a major computational burden of these techniques is due to matrix inversion operations. Applying the iterative FD PN suppression method causes a significant processing delay increase. A resulting increased latency is attributed in part to the intrinsic delay within the FEC operation and in part to the repetition of the computations in the feedback loop. To keep the computational complexity and the latency low, a specific pilot arrangement among the data stream has been proposed. The estimation problem is then solved using a least squares (LS) solution. Nevertheless, achieving consistent results requires solving equations for several sets of pilot symbols, which increases the pilot overhead significantly. This deficiency is related to neglecting of the additive noise in the LS solution. Minimum mean square error (MMSE) solvers, on the other hand, may be used but require a knowledge of the statistical behavior of the PN, which may not be available in practice. Other problems related to the FD PN suppression methods, include, for example, the intrinsic periodicity of the estimated PN that results in a poor estimation performance near the edges of the OFDM symbol. The assumption that PN is a low frequency process may not be verified for some LO implementations.
Time Domain PN Suppression Method
The TD PN suppression method tries to estimate the effect of PN before actual demodulation of samples of the OFDM signal vector. By de-rotating the samples of the OFDM signal vector in the time domain, the OFDM demodulation is then performed on compensated samples. FIG. 3 is a block diagram of an OFDM network implementing a conventional time domain phase noise removal method. On FIG. 3, an OFDM network 1 comprises a transmitter side in which an encoder 2 includes a symbol generator 4 receiving data to be transmitted over the OFDM network 1. The symbol generator 4 applies an M-level modulation (hence “M-ary modulation”) to generate, in the frequency domain, constellation symbols S. The constellation symbols S are applied to an OFDM modulator 6 to generate, in the time domain, a baseband OFDM signal vector xI. The baseband OFDM signal vector xI is then applied to a transmit-side LO 8 that converts the baseband OFDM signal vector xI to a radiofrequency OFDM signal x. The radiofrequency OFDM signal x may be amplified by an amplifier (not shown) before being transmitted by an antenna (not shown) over a channel 10.
The OFDM network 1 also comprises a decoder 12 on a receiver side. The radiofrequency OFDM signal x being received on the channel 10 as a radiofrequency OFDM signal ych on an antenna (not shown). After being amplified by an amplifier (not shown), the radiofrequency OFDM signal ych is applied to a receive-side LO 14 that converts the radiofrequency OFDM signal ych to a baseband OFDM signal vector y. A PN estimation block 16 applies the TD PN estimation method to calculate a PN estimation Ø, using this PN estimation Ø to de-rotate the baseband OFDM signal vector y based on an estimation of the PN, thereby generating a baseband OFDM signal vector ŷ in which at least some of the PN effects have been suppressed. The OFDM signal vector ŷ is applied to an OFDM demodulator 18 that generates, in the frequency domain, samples of the demodulated OFDM signal vector Ŷ. In turn, the samples of the demodulated OFDM signal vector Ŷ are applied to a slicer 20 that generates constellation symbols Ŝ.
The constellation symbols Ŝ reproduce, as much as possible, the constellation symbols S from the encoder 2. However, errors may be present in the constellation symbols Ŝ. An FEC processor 22 may detect and correct data errors present in the constellation symbols Ŝ.
A mathematical derivation of the TD PN suppression method illustrated in FIG. 3 is briefly explained as follows. Referring again to FIG. 4, the end-to-end transmission may be expressed according to equation (2):
                    y        =                                            ψ              tot                        ⁢                          H              m                        ⁢                          1              N                        ⁢                          F              H                        ⁢            S                    +          w                                    (        2        )            
In equation (2), F is a discrete Fourier transform (DFT) matrix of size N and FH is a Hermitian transpose of F. Hm is a circular convolution matrix of the channel 10 with attenuation vector h. ψtot=diag{ejϕ}, wherein diag{X} is a diagonal matrix with the diagonal elements of X. Since Hm is a circulant matrix, it may be diagonalized using the DFT matrix as
      H    m    =            1      N        ⁢          F      H        ⁢    Λ    ⁢                  ⁢          F      .      It may be noted that
      F          -      1        =            1      N        ⁢                  F        H            .      Replacing the diagonalized Hm in equation (2) gives the following equation (3):
                    y        =                                            1              N                        ⁢                          ψ              tot                        ⁢                          F              H                        ⁢            Λ            ⁢                                                  ⁢            S                    +          w                                    (        3        )            
The TD PN suppression method is formulated as finding the diagonal matrix ψ, such that, ideally, ψψtot=IN, wherein IN is the unitary matrix of dimension N. If the desired matrix is found, one may simply let ŷ=ψy, meaning that y is de-rotated, and send ŷ to the OFDM demodulator 18. In that case, the output of the OFDM demodulator 18 becomes Ŷ=ΛS+W.
The diagonal matrix ψ may be estimated on the basis of its diagonal elements. Thus, Φ is defined as the vector of the diagonal elements of the diagonal matrix ψ. The TD PN suppression method estimates the vector Φ using the following two components.
Firstly, a set of d basis vectors is used, wherein each basis vi is a N dimensional vector. This gives that the total basis vectors are placed in the matrix V=[v1, . . . , vd].
Secondly, the vector of d coefficients corresponding to the basis vectors (1 scaler coefficient for each basis vector) is used. The coefficient vector is shown with γ.
The desired vector Φ is written as Φ=Vγ. Thus, ŷ=ψy=YmVγ, wherein Ym is a diagonal matrix with the vector y on its diagonal. The OFDM demodulator 18 acts as an FFT operator, in the sense that Ŷ=Fŷ. This may also be written as Ŷ=FYmVγ. By letting M=FYmV, equation (4) may be formed:Mγ=ΛS+{tilde over (W)}→S=Λ−1Mγ+{tilde over (W)},  (4)
In equation (4), {tilde over (W)} is a vector of AWGN. Estimating the PN samples is equivalent to finding the matrix V and vector γ. The basis vectors are assumed to be chosen and fixed; this is a mild assumption as there exists some basis sets that may be readily used. As an example, a discrete cosine transform (DCT) basis may be used. Thus, γ may be estimated from equation (4). There is in equation (4) a set of N equations. However, the elements of the constellation symbols S are not (all) known so all the N equations cannot be used. Instead the equations corresponding to the pilot locations are used. Assuming there are L pilots, the set of L equations is written as expressed in equation (5):Sp=[Λ−1M]pγ+{tilde over (W)}p  (5)
All the elements in equation (5) are known except for γ, which is the desired vector. To find γ, the LS solution is used, which gives {circumflex over (γ)}=[[Λ−1M]p]†Sp.
Comparing the FD and TN PN Suppression Methods
The TD PN suppression method has received less attention when compared to the FD PN suppression method. Nevertheless, the TD PN suppression method has a superior estimation performance when compared the FD PN suppression method, when both methods are used with equal pilot symbol overhead and comparable computational complexity and latency.
FIG. 4 is a graph comparing a performance of conventional frequency domain and time domain phase removal methods. For further verifications, the mean squared error (MSE) of the PN estimation between the TD PN suppression method and the FD PN suppression method, which are considered the state-of-the-art in the PN suppression in OFDM system, are compared in FIG. 4. On a graph 30, the FD and TD PN suppression methods are compared when using a 256-QAM constellation and a SNR of 32 dB. MSE is calculated as 20×log10|ϕ−{circumflex over (ϕ)}|, where ϕ is the actual PN and ϕ is the estimated PN at a given SC. The number of OFDM symbols is 50, and each symbol contains 3300 active SCs. The FFT size is 4096. These numbers are in line with the OFDM requirements mentioned in the release 15 of the 3rd Generation Partnership Project (3GPP) specification when the channel bandwidth is 400 MHz. In both cases, the pilot symbol overhead is 1%.
On the graph 30, a MSE curve 32 represents the performance when the FD PN suppression method is limited to CPE removal. A MSE curve 34 represents the estimation performance when l equal to 4 frequency components are used. This involves taking the inverse of a matrix with dimension (2l+1)×(2l+1), which is 9×9 in this case. A MSE curve 36 represents the performance when the TD PN suppression method is used with a number d of basis vectors is set to 8.
Comparing MSE curves 34 and 36, the performance of the TD PN suppression method, with the same pilot symbol overhead, is almost 5 dB better than that of the FD PN suppression method. It would be possible to improve the performance of the FD PN suppression method to the level of the TD PN suppression method, but that could only be done at the cost of a significant increase of the pilot symbol overhead. This is due to the fact that when using the LS solution, which is more practical than using the MMSE solution, the effect of the additive noise and of the ICI terms (in equation (1)), other than those that are being estimated are neglected. To avoid this problem, more pilot symbols and more equations (for the same set of ICI terms in equation (1)) would need to be used. The final result would then be the average of the answers of equation (1). Compared to the TD PN suppression method, where the estimation problem is solved using the LS solution just once, the FD PN suppression method clearly requires more overhead and greater computational complexity.
Although TD PN suppression method is an attractive solution to suppress the PN in the OFDM systems, its performance is no longer satisfactory when the OFDM network 1 relies on larger constellation sizes. When using high order modulations, for example 256-QAM and higher, the TD PN suppression method with a reasonable overhead and a small set of basis vectors fails to meet the PN estimation performance requirements. Satisfactory performance may only be obtained by increasing the pilot symbol overhead and the number of applied basis vectors. The former contradicts with the actual intent of using the large constellations, which is to reach high spectral efficiency, while the latter significantly increases the computational complexity. The high computational complexity is due to the requirement for a higher number of basis vectors used in the estimation process. A matrix inversion with the dimension equal to the number of basis vectors is necessary and the matrix inversion, a complexity of this operation increasing with a cubic order of the number of basis vectors.
FIG. 5 is a graph comparing a performance of the PN estimation using the time domain method for different overhead and number of basis vectors combinations. A graph 40, produced using the same simulation parameters as those used to produce the graph 30, illustrates simulation results on a MSE curve 42 for the TD PN suppression method, with a number of basis vectors of 10, which translates to a 10×10 matrix inversion, and 1% pilot symbol overhead. The performance of MSE curve 42 is not sufficient for a 1024-QAM constellation. A MSE curve 44 shows simulation results for the TD PN suppression method with a number of basis vectors of 40, which translates to a 40×40 matrix inversion, and 10% pilot symbol overhead. The MSE curve 44 meets the PN estimation requirements for a 1024-QAM constellation. The simulation results confirm that reaching the PN estimation performance that are required when large constellations are used necessitates a large overhead ratio and complex matrix inversion operations. Unfortunately, a pilot symbol overhead as large as 10% is excessive from a spectral efficiency standpoint, and performing a 40×40 matrix inversion for each OFDM symbol is not realistic.
The LOs are indispensable parts of every digital communication device and the PN is presence is almost all types of LOs. Consequently, there is a need for improved PN suppression techniques that overcome the above-described inconveniences.